Chapter 1 Length Structures: Path Metric Spaces
Introduction
In classical Riemannian geometry, one begins with a C^ manifold X and then studies smooth, positive-definite sections g of the bundle S^T*X. In order to introduce the fundamental notions of covariant derivative and curvature (cf. [Grl-Kl-Mey] or [MilnorjMT, Ch. 2), use is made only of the differentiability of g and not of its positivity, as illustrated by Lorentzian geometry in general relativity By contrast, the concepts of the length of curves in X and of the geodesic distance associated with the metric g rely only on the fact that g gives rise to a family of continuous norms on the tangent spaces T^X of X. We will study the associated notions of length and distance for their own sake
A. Length structures
1.1. Definition: The dilatation of a mapping / between metric spaces X, Y is the (possibly infinite) number where "oJ" stands for the metrics (distances) distx in X and disty in Y. The localdilatation of / at 2:is the number diU(/) = limdil(/|B(.,.)).
Metric Structures for Riemannmn and Non-Riemannian Spaces
A map / is called Lipschitz if dil(/) < OD; it is called X-Lipschitz if dil(/) < A, in which case, the infimal such Ais called the Lipschitz constant of /.
If / is a Lipschitz mapping of an interval [a, b] into X, then the function 11—^ d\\t{f) is measurable
1.2. Definition: The length of a Lipschitz map / : [a,6] —^X is the number
If / is merely continuous, we can define £{f) as the supremum of all sums of the form Yl'i^od{f{U), fiU+i)) where a = to < ti < ---< tn^i = 6 is a finite partition of [a,b].
If (/? is a homeomorphism of a closed interval /' onto I = [a,6], then i satisfies i{foif) = i{f), as follows from the fact that (f is strictly monotone (invariance under change of parameter).
The two definitions of i{f) stated above are equivalent when / is absolutely continuous (cf. [Rinow], p. 106). This fact permits us to define £{f) as the integral of the local dilatation when / is Lipschitz and to set i{f oif) = £{f) for each homeomorphism (^ of 7 onto /'. More generally,
1.3. Definition: A length structure on a set X consists of a family C{I) of mappings /: / — ^ X for each interval I and a map i of C = [jC{I) into R having the following properties:
(a) Positivity: We have £(/) > 0 for each f e C, and £{f) = 0 if and only if / is constant (we assume of course that the constant functions belong to C)
(6) Restriction, juxtaposition: If / C J, then the restriction to / of any member of C{J) is contained in C{I). If / G C([a,6]) and g EC{[b,c]), then the function h obtained by juxtaposition of / and g lies in C{[a,c]) and i{h) = £{f) + ({g).
(c) Invariance under change of parameter: If (/? is a homeomorphism from / onto J and if / G C(J), then f o ip e C{I) and eifoip) = e{f).
(d) Continuity: For each / = [a,6], the map 11-> £{f\[a,t]) is continuous
Using conditions (a), (6), and (c), we can define a pseudo-metric di on X called the length metric by setting d,{x,y) = mi{i{f) :feC,x,ye im(/)}.
Length Structures: Path Metric Spaces
As usual, this pseudo-metric induces a topology on X. It is common to define i{f) = oc when the map f : I ^^ X is not contained in C(/)
1.4. Examples:
(a) A metric space (X, d) is equipped with a canonical length structure: The set C consists of all continuous mappings from intervals into X, and the function £ is defined as in 1.2 above The resulting structure is called the metric length structure of (X,d); in general, however, the length metric di differs from d, and their corresponding topologies may also be distinct
(64.) Tits-like metrics and snowflakes: Consider R^ equipped with polar coordinates (r,5), where r G [0,oo) and s G 5^"-^, the unit sphere in R^. Define
d{xi,X2) = In -r2| -f r||5i -S2p/^,
where Xi = (r^,Sj) G R'^, i = 1,2, ||si — 52II denotes the Euclidean distance on S^~^ C R", and r = min{ri, r2}. This d gives rise to the usual topology on R^, but
de{x,,x,) = l l^^-^^l f'^^Z'^
and so (R'^, di) becomes the disjoint union of the Euclidean rays [0,00) x 5 for all s € 5"~^, all glued together at the origin only. In particular, the unit sphere S^~^ C R^ is discrete with respect to de. Metrics of this type naturally appear on (the ideal boundaries of) manifolds with nonpositive sectional curvatures and are collectively referred to as Tits metrics (cf [Ba-Gr-Sch]).
An analog of the metric de can be constructed on the subset of Euclidean 3-space consisting of the straight cone X C R^ over the Koch snowflake 5 C R^ (Here, the snowflake isthe base of the cone and plays the role of the sphere S^~^ in the Tits-like example above). The only curves in X having finite Euclidean length are those contained in the (straight) generating lines of the cone, and so these lines are disjoint with respect to di away from the vertex (compare [Rinow], p 117, and Appendix B_^ of this book)
In general, the metrics d, di always satisfy the inequality d < di, so that their corresponding topologies coincide if and only if for each x e X and £ > 0, there exists a ^-neighborhood of x in which each point is connected to Xby a curve of length at most e.
Metric Structures for Riemannian and Non-Riemannian Spaces
(c) If X is a manifold, then any Riemannian or Finslerian structure on X naturally gives rise to a length structure: One proceeds as in 1.2, noting that when / is differentiable, its local dilatation at a point x equals the norm of its derivative at x.
(d) Induced length structures: If X is equipped with a length structure and (^ is a map from a set Y into X, then we obtain a length structure on Y by setting for each f:I-^Y.
(e) First exposure to Carnot-Caratheodory spaces. We can associate a length structure on a Riemannian manifold (V,g) with any tangent subbundle E C TV by defining the length of a curve c to be its usual Riemannian length if c is absolutely continuous and its tangent vector lies within E at a.e point, and by setting ^(c) = oo otherwise If E is integrable, then the topology defined by d(_ is none other than the leaf topology. The case of nonintegrable E is of great interest.
A basic example of the latter structure is provided by the 3-dimensional Heisenberg group M^ of matrices of the form
1. Length Structures: Path Metric Spaces 5
equipped with a left-invariant metric. The quotient of H^ by its center C (isomorphic to R) defines a Riemannian fibration (see [Ber-Gau-Maz], Ch. 1) of H^ over the Euchdean plane M^/C c^ R^ rpj^^ subbundle E then consists of the horizontal subbundle of this fibration, which coincides with the kernel of the 1-form dz — xdy.
1.5. Suppose X is equipped with a length structure ^, and let i be the length structure defined by the metric d£. The following criterion, which is nothing more than an axiomatic version of the classical properties of the lengths of curves in metric spaces, describes when these two structures are identical.
1.6. Proposition: //, for each interval I, the function i is lower semicontinuous on C{I) with respect to the compact-open topology, then £= i.
Proof. By 1.3(d), the function t \-^ ^(/|[a,t]) is uniformly continuous on / = [a, b]. For each £ > 0, there exists 77 > 0 such that if |t — t'| < r/, then de{m,f{t'))<s.
Let a = to < ti < '" < tn-f1= 6 be a partition of / having increments no larger than rj. For each integer i between 0 and n, there exists a map gi in C{[ti,ti^i]) having the same values as / at ti,ti^i such that i{gi)<de{f{ti)J{ti^i))-hs/n.
By juxtaposing the gi, we obtain a curve h^ satisfying n n
i{h,) = Y,i{9i) < YldeifiU), fiti+i)) + e < i{f) + e
and such that for each t G/, we have d(,{h^{t),f{t)) < 3e.
From the hypothesis that i is lower semicontinuous, it follows that i{f)<\imini£{h,)<i{f),
whereas the opposite inequality is an immediate consequence of the definition of L
Remark: If i is the length structure associated with a metric d, then the same argument as above shows that £ = i, using the semicontinuity of length with respect to d (cf. 1.2 and [Choq], p. 137). In other words, by following the sequence of constructions (X,d), a metric metric length (X, d^), a new metric space —y structure £ —> associated with the on X length structure.
Structures for Riemannian and Non-Riemannian Spaces
we obtain the same length structure Nevertheless, we again emphasize that £7^£ in general.
1.6^ Locality of the length structure. If two length structures agree on some open subsets covering X, then they are obviously equal. Conversely, if we are given a covering of X by open subsets Xi for i G /, together with length structures £{ on the Xi which are compatible on the intersections Xi r\Xj for all i,j £ I, then there obviously exists a (unique) length structure on X that restricts to £i on each Xi. (In other words, the length structures comprise a sheafover X.) On the other hand, metrics on X are not local (they form only a presheaf over X), but they can be localized as follows: Given a metric d on X, we consider all metrics d' that are locallymajorized by d. This means that for each x EX, there exists a neighborhood Y^ C X ofx such that d|y^ > d'ly^. Now take the supremum of all these d' and call it dm- (Note that the supremum of a bounded family of metrics is again a metric. In general, this supremum may be infinite at some pairs of points in X, but otherwise it looks like a metric.) Clearly dm ^ di in any metric space (X, d); if (X, d) is complete^ then dm =" d£, as a trivial argument shows (see Section 1.8 below).
B. Path metric spaces
1.7. Definition: A metric space (X^d) is a path metric space if the distance between each pair of points equals the infimum of the lengths of curves joining the points (i.e., ii d= di).
Examples: Note that, according to this definition the Euclidean plane is a path metric space, but the plane with a segment removed is not.
The n-sphere S^ is not a path metric space when equipped with the metric induced by that of R"^"^^, but it is a path metric space for the geodesic metric by Proposition 1.6
Path metric spaces admit the following simple characterization
1.8. Theorem: Thefollowing properties of ametric space (X, d) areequiv-
alent:
1. For arbitrary points x,y ^ X and e > 0, there is az such that sup(d(x, z), d{z, y)) < - d{x, y) + e,
2. For arbitrary x^y EX and ri,r2 > 0 with ri + r2 < d{x^y), we have d{B{x, ri), B{y, r2)) < d{x, y) - ri - r2, for
d{B^,B2)= inf dix',y'). y' e B2
Every path metric space has these properties, and conversely, if (X, d) is complete and satisfies (1) or (2), then it is apath metric space.
Proof. Let (X, d) be a complete metric space satisfying condition (1), and set 5 = d{x,y). Given a sequence (sk) of positive numbers, there is a point 2:1/2such that max(d(x, 2:1/2), d{zi/2, y)) £ 6/2-\-£iS/2, and points 2:1/4,2:3/4 for which each ofthe distances d(x, 2:1/4),(^(2:1/4,2:1/2), d{zi^2i ^3/4)7<^(^3/4) v) are less than
l/2{S/2 + EiS/2) + S2{d/2 + siS/2), etc
By choosing the sequence (sk) so that ^i^Sk < 00, we can define a map / from the dyadic rationals in [0,1] into X satisfying If (X,d) is complete, then this map extends to the entire interval [0,1] Since we can choose the Sk so that the product 11(1 •^" ^A;) is arbitrarily close to 1, we obtain curves whose lengths tend to 6= d(x,?/), which proves the last assertion.
The implication (1) => (2) is proven in the same way, whereas (2) =^ (1) and the assertion that a path metric space satisfies (1),(2) are trivial.
Path metric spaces enjoy some of the same geometric properties as Riemannian manifolds.
l.Sbis. Property: If (X,d) is a path metric space, and if / is a map of X into a metric space Y, then the dilatation of / obviously equals the supremum of its local dilatation, i.e., dil(/) = sup^^;^dila;(/). Note that
Metric Structures for Riemannian and Non-Riemannian Spaces
if X and Y are Riemannian manifolds, and if / is differentiable, then the differential Dfx'. T^X -^ Tf(^j.)Y satisfies dila:(/) = ||Z)/a;||.
1.8bis-j Kobayashi metrics. Let A be a path metric space and let X be an arbitrary (say, topological) space with a distinguished set of maps f:A—^X. Consider all metrics d' on X for which these / have dil(/) < 1, i.e., for which the mappings are (nonstrictly) distance decreasing, and define dx as the supremum of the metrics d' on X. Here, it is convenient to admit degenerate metrics d' (in the sense that d'[x^y) — 0 for perhaps some X 7^ ?/), so that dK may itself be degenerate. In fact, this dx is a (possibly degenerate) path metric by the property above.
In the classical example, due to Kobayashi, A is the unit open disk equipped with the Poincare metric (i.e., the hyperbolic plane), X is a complex analytic space, and the collection of distinguished maps consists of all holomorphic mappings A —> X The usefulness of this metric is based on the Schwarz lemma (and its various generalizations), which implies that dK is nondegenerate for many X. Such X are said to be (Kobayashi) hyperbolic. For example, the disk A is itself hyperbolic since dx in this case equals the Poincare metric (following from the fact that every holomorphic map A —)•A is distance decreasing with respect to the Poincare metric, a consequence of the classical Schwarz-Ahlfors lemma). The basic features of the Kobayashi metric and hyperbolicity do not depend on the integrability of the implied (almost) complex structure of X and therefore extend to all almost complex manifolds X (via the theory of pseudo-holomorphic curves in X, cf. [McD-Sal]). For example, hyperbolicity is stable under small (possibly singular) perturbations of almost complex structures on compact manifolds and (suitably defined) singular almost complex spaces (compare [Kobay], [Brody], [Krug-0ver]).
There is also a real analog of Kobayashi hyperbolicity, in which X is a Riemannian manifold, A is as above, and the set of distinguished maps consists of all conformal, globally area minimizing mappings A —^X. In this case, hyperbolicity ofX isequivalent to ^-hyperbolicity (see (e) in1.19^ _ below) under mild restrictions on X, which are satisfied, for example, if X is the universal cover of a compact manifold (In fact, X does not have to be a manifold here — it can be a rather singular space, e.g., a simplicial polyhedron as in 1.1S^, see [GroJnG, [GroJHMGA-)
1.9. Definition: A minimizing geodesic in a path metric space {X,d) is any curve f: I -^ X such that d{f{t)J{t')) = \t- f\ for each t,t' e L A geodesic in X is any curve f: I -^ X whose restriction to any sufficiently small subinterval in / is a minimizing geodesic.
In this connection, we have the following :
Hopf—Rinow theorem. If (X, d) is a complete, locallycompact path metric space, then
1. Closed balls are com,pact, or, equivalently, each hounded, closed domain is compact.
2. Each pair ofpoints can hejoined hy a minimizing geodesic.
Before turning to the proof of the theorem, we observe that if (X, d) is a complete, locally compact metric space, then there are many noncompact balls for the metric d' = inf(l,d).
1.11 Compactness of closed balls. Note that if a is a point in X, then the ball B{a^r) is by hypothesis compact if r is sufficiently small. We will first show that if B{a^r) is compact for all r in an interval [0,/o), then B{a^p) is compact as well
Let {xn) be a sequence of points in B{a^p). We may suppose that the distances o?(a,Xn) tend to p; otherwise, there is a ball B{a^r) with r < p containing infinitely many of the Xn and thus a limit point of the sequence. Let (sp) be a sequence of positive real numbers tending to zero By applying property (2) of Theorem 1.8, we find that for each p, there exists an integer n{p) such that for each n > n{p), there is a point y^ satisfying
y^ e B{a, p- 2£p) and d{xn, y^) <Sp.
For each p, the sequence (t/^) lies within a compact set; by a diagonal argument (or since the product of compact sets is compact), it follows that there is a sequence of integers (uk) such that the subsequence (^/n^) converges for all p. The sequence (xn^), which is the uniform limit of the {Vrik)^ is a Cauchy sequence and therefore converges by completeness of X. By the preceding remarks, the supremum of the r for which S(a, r) is compact is infinite: if instead it equalled /?< oc, then we could find p' > p such that B{a,p) would be compact, by using a finite covering of the sphere 5(a, p) by compact balls.
1.12. Existence of a minimizing geodesic joining two arbitrary points
We first consider the case when X is compact.
Lemma: // {X, d) is a compact path metric space and a, 6€ X, then there exists a curve of length d{a^b)joining a and b.
Metric Structures for Riemannian and Non-Riemannian Spaces
Proof. It suffices to consider curves /: [0,1] -^ X which are parametrized by arc length. From the definition of path metric spaces, if follows that for each positive integer n, there exists such a curve fn joining a to 6and having length less than (i(a,6) -h 1/n The set of fn is therefore equicontinuous, and by Ascoli's theorem, there exists a subsequence fn,, that converges uniformly to a curve / : [0,1]^ X. Since the length function £ is lower semicontinuous, we have
i{f)<\imM£{fn,) = d{a,b).
In the case of a complete, locally compact, but noncompact path metric space, it suffices to note that the images of the curves fn chosen in the preceding paragraph lie within the compact ball B{a,2d{a,b)).
1.13. Remarks: (a) In the case of Riemannian manifolds, this proof fulfills the promise made in the introduction that use would be made only of the associated length structure.
(6) The equicontinuity argument of Lemma 1.12 also shows that in a compact path metric space, everyfree homotopy classis represented bya lengthminimizing curve^ and that the minimizing curves are geodesies. Moreover, if X is a manifold, then for each real r, there is only a finite number of homotopy classes represented by curves of length less than r (again, it suffices to use Ascoli's theorem and the fact that the homotopy classes are open subsets of C^{S^,X); cf. [Dieu], p. 188). These results also hold for homotopy classes of curves based at a point x e X and geodesies based at X (but not necessarily smooth at x) and will play a key role, particularly in Chapter 5.
Examples of path metric spaces
1.14. Riemannian manifolds with boundary and subsets of R^ with smooth boundary. Let X be a domain in R^ with smooth boundary, equipped with the metric and length structure induced by that of R^, and let / be the identity map (X,induced metric)^ (X,induced length metric).
It is easy to see that if the boundary of X is smooth, then dilx(/) = 1 for each XGX, and that dil(/) = 1 if and only if X is convex.
Distortion.^: More generally, let X be a subset of a path metric space A and let distort(X) denote the dilatation of the identity map f: X —^ X with respect to the two induced metrics, i.e., .. ^ ^/v\ (length distort(X)=sup-^^^^dist)IX.
Our first observation is the following:
(a) Let X be a compact subset of W^. //distort (X) < | (which means that every two points in X that lie within a Euclidean distance of d from one another can bejoined by a curve in X of length < dn/2), then X is simply connected.
Proof. Toprove this assertion, weargue by contradiction Suppose TTI(X) 7^ 0 and let a be a nontrivial homotopy class in which there exists a curve of minimal length among all homotopically nontrivial loops (the existence of such a is guaranteed by Remark 1.13(b)) Let Y be the image of c and g :Y -^ Y the identity map of the space Y with its induced length structure We claim that dil(^) = dil(/|y)
To prove the claim, let 1/1, yo be two points of Y and fix a parametrization of Y by arc length, i.e., a map c: [0,£]^ Y such that c(0) = y^ = c{£) and 2/1= c{d) for some d E [0, £] such that d < i —d. Then c|[o,d] is the shortest path joining yo to yi in X. Indeed, if there were a strictly shorter path c' from yo to t/i, then the two loops obtained by adjoining c' and the two parts of c defined by the parameters 0 and d would be strictly shorter than c. Since their product is homotopic to c, however, we could conclude that one of the two is not homotopic to 0 in X, which contradicts the minimality of c. Since the path c|[o,d] lies within y, it follows that d is the distance from yo to yi for the length metrics of X and Y, and that
dil(2/o,i/i)(^) =dil(^o,2/i)(/)-
Metric Structures for Riemannian and Non-Riemannian Spaces
Thus, we have dil(^) < 7r/2. Extend c to a periodic function on R and set r{s) = d{c{s),c{s-\-£/2)), so that the inequaHty r{s) > £/2dil{g) holds Set u{s) = {c{s + £/2) — c{s))/r{s). The curve u is differentiable almost everywhere, and its image lies within the unit sphere of R^ Moreover, u{s -h^/2) = —u{s), so that the length of u is at least 27r. Thus,
du ds
4dil(g)y e ) ' and so £{u) < 4dil(^) < 27r, which is the desired contradiction
Remark: If di[{d) = 7r/2, and if X is not simply connected, then X contains a round circle (6) //distort(X) < 7T/2y/2, then X is contractible.
Proof^ The idea is to homotopy-retract R" to X by following the flow of a suitable vector field d which plays the role of — grad d{y) for the distance function d\ y \^ d\st{y^X) = inf^^^x \\y— ^||- In general, the function d is nonsmooth, even on the complement R^ \ X. Nonsmoothness at a point y ^ W^ \X \s due to the fact that the sphere Sy~^ at y of radius d{y) can meet X at several points. These points essentially realize the above infimum, since the open ball bounded by Sy~^ does not intersect X, while the set XnSy'^ is nonempty and d{y) = \\x-y\\ for all x G XnSy~'^. Now weobserve that the normal projection X — ^ ^y~^ isdistance-decreasing and thus (length dist)j^(a:i,X2) > (length dist)^n-i (2:1,3:2) for all pairs of points in the intersection X DSy~^. It follows that the latter distance is bounded by 7rd{y)/2^ and since we assume the strict inequality distort(X) < 7r/2v^, the distance above is bounded by S7rd{y)/2 for some S < 1 independent of y. Consequently, the intersection X n Sy~^ is strictly contained in a hemisphere, or, in other words, there exists a unit vector dy at every y G R^ \ X such that
dy\\x-y\\<-e<0 for all x GX 0 S^-\ (*) where dy\\x — y\\ denotes the dyderivative of the (distance) function y t-^ \\x— y\\. In fact, one can take dy to be the vector which points towards center of the minimal spherical cap in Sy~^ containing X fi Sy~^, thus obtaining a (Borel) measurable vector field y ^-^ dy satisfying (*). Finally, we can easily smooth this vector field, so that the resulting (now smooth!) unit vector field, say y ^-^dy on W^\ X, satisfies (*) (possibly with a smaller £ > 0) as well Clearly, every forward orbit of such a field
converges to a point in X, and sotheflowgenerated by d eventually retracts M^ to X.
Remark^: A sharper result is proved in Appendix A, where one allows distort(X) < l-\-an for some specific a^ > 1 —n/2y/2. Furthermore, there are many examples of (necessarily contractible) subsets X with arbitrarily large (even infinite) distortion for which X nSy~^ is still strictly contained in an open hemisphere for each y e W^ \ X and to which our argument applies. On the other hand, we do not know the precise value of an for which distort(J^) < 1-f a^^ necessarily implies that X is contractible
Exercise^: Construct a closed, convex surface X in R^ with distort(X) < 7r/2. (Compare Appendix A.)
Problem^: Given a topological space X, evaluate the infimum of all distortions induced by embeddings X^ M"" or of those distortions induced by embeddings which lie in a fixed isotopy class. The first interesting case arises when X is the circle and we minimize the distortion for X knotted in M^ in a prescribed way (compare [Gro]HED» [O'HarajEK)-
Remark^: The geometry ofsubsets X C R^ satisfying distort(X) < l-f-a^ can be rather complicated, even for small a^^ > 0 For example, there are simple smooth arcs in R^ with arbitrarily small (i.e. close to 1) distortion which have an arbitrarily large turn of the tangent direction. To see this, consider diffeomorphisms T^: R^ —» R^ with the following properties: Each Ti fixes the complement of the disk of radius 2""^ around the origin and isometrically maps the disk of radius 2~^~-^ into itself by rotating it by a small angle a > 0.
Clearly, one can choose the Ti so that they and their inverses are (1+6:)-
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dropped into a low hollow, and on top of the next small lift in the plains he rode into three riders, one of whom was a woman.
Rock had keen eyes. Moreover, since that meeting with Elmer Duffy he was acutely conscious of his newly acquired identity. Thus he marked instantly the brands of the horses. Two were Maltese Cross stock, the other, bestridden by a youth of twenty or less, carried Nona Parke’s brand on his left shoulder. His rider was a blueeyed slender boy, with a smile that showed fine white teeth when he laid his eyes on Rock.
“Hello, Doc, old boy,” he said. “How’s the ranch an’ the family and everythin’?”
“Same as usual,” Rock answered genially. “What you expect?”
They had reined up, facing each other. The second man nodded and grunted a brief, “Howdy.” The girl stared at Rock with frank interest, as he lifted his hat. Her expression wasn’t lost on him. He wondered if he were expected to know her well, in his assumed identity. In the same breath he wondered if a more complete contrast to Nona Parke could have materialized out of those silent plains. She was a very beautiful creature, indeed. It was hot, and she had taken off her hat to fan her face. Her hair was a tawny yellow. A perfect mouth with a dimple at one corner fitted in a face that would have been uncommon anywhere. Curiously, with that yellow hair she had black eyebrows and eyelashes. And her eyes were the deep blue, almost purple, of mountains far on the horizon. To complete the picture more effectually her split riding skirt was of green corduroy, and she sat atop of a saddle that was a masterpiece of hand-carved leather, with hammered-silver trimmings. It was not the first time Rock had seen the daughters of cattle kings heralding their rank by the elaborate beauty of their gear. He made a lightning guess at her identity and wondered why she was there, riding on roundup. She seemed to know him, too. There was a curious sort of expectancy about her that Rock wondered at.
However, he took all this in at a glance, in a breath. He said to the boy on the Parke horse:
“Where’s the outfit?”
“Back on White Springs, a coupla miles. You might as well come along to camp with us, Doc. It’s time to eat, an’ you’re a long way from home.”
“Guess I will.” Rock was indeed ready to approach any chuck wagon thankfully. It was eleven, and he had breakfasted at five.
They swung their horses away in a lope, four abreast. What the deuce was this Parke rider’s name, Rock wondered? He should have been primed for this. Nona might have told him he would possibly come across the Maltese Cross round-up. This must be her “rep.”
And he was likewise unprepared for the girl’s direct attack. Rock rode on the outside, the girl next. She looked at him sidewise and said without a smile, with even a trace of resentment:
“You must be awful busy these days. You haven’t wandered around our way for over two weeks.”
“I’m working for a boss that don’t believe in holidays,” he parried.
“I’d pick an easier boss,” she said. “Nona never lets the grass grow under anybody’s feet, that I’ve noticed. Sometimes I wish I had some of her energetic style.”
“If you’re suffering from lack of ambish,” Rock said, merely to make conversation, “how’d you get so far from home on a hot day?”
“Oh, Buck was in at the home ranch yesterday, and I rode back with him. Took a notion to see the round-up. I think I’ll go home this afternoon.”
“Say, where’d you get that ridin’ rig, Doc?” the young man asked. He craned his neck, staring with real admiration, and again Rock felt himself involved in a mesh of pretense which almost tempted him to proclaim himself. But that, too, he evaded slightly. He did have a good riding rig. It hadn’t occurred to him that it might occasion comment. But this youth, of course, knew Doc Martin’s accustomed gear probably as well as he knew his own. Naturally he would be curious.
“Made a trade with a fellow the other day.”
Rock registered a mental note to cache Martin’s saddle, bridle, and spurs as soon as he got home.
“I bet you gave him plenty to boot,” the boy said anxiously. “You always were lucky. He musta been broke an’ needed the mazuma.”
“I expect he was,” Rock agreed.
Again the girl’s lips parted to speak, and again the boy interrupted. Rock out of one corner of his eye detected a shade of annoyance cross her alluring face. He wondered.
“How’s Nona an’ the kid?”
“Fine,” Rock informed him. “I left her riding down to Vieux’s after that dark-complected nurse girl.”
“Are you going back home to-night?” the girl asked abruptly.
“I’d tell a man,” Rock said. “As soon as I do business with the chuck pile, I’m riding. I’m supposed to be back by three, and I’ll certainly have to burn the earth to make it.”
“You won’t lose your job if you don’t.”
“Well, if I do, I know where I can get another one,” Rock said lightly. “But I aim to be on time.”
“Him lose his job!” the TL rider scoffed. “You couldn’t pry him lose from that job with a crowbar. Now don’t shoot,” he begged in mock fear. “You know you got a snap, compared to ridin’ round-up with the Maltese Cross—or any other gosh-danged cow outfit. I’m goin’ to put up a powerful strong talk to Nona to send you on beef round-up this fall an’ let me be ranch boss for a rest.”
“You got my permission,” Rock said a little tartly. These personalities irked him. “I’ll be tickled to death if you do.”
He didn’t know what there was in his words, or tone, perhaps, to make the boy stare at him doubtfully, and the yellow-haired girl to smile with a knowing twinkle in her eyes, as if she shared some secret understanding with him.
By then they were loping swiftly into a saucerlike depression in the plains, in the midst of which a large day herd grazed under the eye of four riders, and the saddle bunch was a compact mass by the round-up tents.
Rock left his horse standing on the reins. The others turned their mounts loose. The Cross riders were squatted about the chuck wagon in tailor-fashion attitudes, loaded plates in their laps. Rock followed the other three to the pile of dishes beside the row of Dutch ovens in the cook’s domain. Some of the men looked up, nodded and called him by name. And, as Rock turned the end of the wagon, he came face to face with a man holding a cup of coffee in one hand—a man who stared at him with a queer, bright glint in a pair of agate-gray eyes, a look on his face which Rock interpreted as sheer incredulity.
He was a tall man, a well-built, good-looking individual, somewhat past thirty, Rock guessed. His clothing was rather better than the average range man wore. Neither his size nor his looks nor his dress escaped Rock’s scrutiny, but he was chiefly struck by that momentary expression.
And the fellow knew Rock. He grunted: “Hello, Martin.”
“Hello,” Rock said indifferently. Then, as much on impulse as with a definite purpose, he continued with a slight grin: “You seem kinda surprised to see me.”
Again that bright glint in the eyes, and a flash of color surged up under the tan, as if the words stirred him. Rock didn’t stop to pry into that peculiar manifestation of a disturbed ego. He was hungry. Also, he was sensible and reasonably cautious. He felt some undercurrent of feeling that had to do with Doc Martin. Between the vivacious blonde and this brow-wrinkling stockman, Rock surmised that posing as Doc could easily involve him in far more than he had bargained for.
So he filled his plate and busied himself with his food. No one tarried to converse. As each rider finished eating, he arose, roped a fresh horse out of the remuda, and saddled. The girl and the other two riders ate in silence. From the corner of one eye Rock could see the girl occasionally glance at him, as if she were curious or tentatively expectant. He couldn’t tell what was in her mind. He was going it blind. He didn’t know a soul whom he was supposed to know. That
amused him a little—troubled him a little. The quicker he got on his way the better. He had got a little information out of this visit, though. He heard one of the riders address the big, well-dressed man as “Buck.” He heard him issue crisp orders about relieving the day herders. Old Uncle Bill Sayre’s words floated through his mind: “Buck Walters is young, ambitious and high-handed with men an’ fond of women. He dresses flash. A smart cowman.”
That was Buck Walters, the range-functioning executor of the Maltese Cross estate. And there was some distaste in Buck Walters for Doc Martin. More wheels within wheels. Rock wondered if this tawny-haired girl could be the daughter of the deceased Snell. Probably. That didn’t matter. But it might matter a good deal to him if there was any occasion for bad blood between Walters and the dead man into whose boots he, Rock, had stepped.
He finished and rose.
“Well, people,” said Rock, “I’ll be like the beggar, eat and run. I have a long way to go.”
“Tell Nona to ride over to see me,” the girl said politely, but with no particular warmth. “I’ll be at the ranch most of the summer.”
“Sure,” Rock said laconically. “So long.”
He was a trifle relieved when he got clear of that camp. He had plenty of food for thought, as he covered the miles between White Springs and the Marias. Stepping out of his own boots into those of a dead man seemed to have potential complications. When Rock pulled up on the brink of the valley, he had just about made up his mind that he would be himself. Or, he reflected, he could turn his back on Nona Parke and the TL, and the curious atmosphere of mystery that seemed to envelope that ranch on the Marias. He was a capable stock hand. He could probably work for the Maltese Cross and learn all he wanted to know under his own name. Why burden himself with a dead man’s feud, even if the dead man might have been his brother?
As far as Nona Parke went, one rider was as good as another to her. And Rock had no intention of remaining always merely a good stock hand. Other men had started at the bottom and gained
independence. No reason why he should not do the same. Land and cattle were substantial possessions. Cattle could be bought. From a small nucleus they grew and multiplied. Land could be had here in the Northwest for the taking. Why should he commit himself to a dead man’s feuds and a haughty young woman’s personal interests? For a monthly wage? He could get that anywhere. He could probably go to work for the Maltese Cross, without question and in his own identity.
Rock, looking from the high rim down on the silver band of the Marias, on the weather-bleached log buildings, asked himself why he should not ride this range and fulfill his promise to an uneasy man in Texas in his own fashion? Why shouldn’t he work for some outfit where there were neither women to complicate life, nor enemies save such as he might make for himself?
The answer to that, he decided at last, must be that one job was as good as another, and that somehow, for all her passionate independence, Nona Parke needed him. There was a peculiar persuasiveness about that imperious young woman. Rock could easily understand why men fell in love with her, desired her greatly, and were moved to serve her if they could. She seemed to generate that sort of impulse in a man’s breast. Rock felt it; knew he felt it, without any trace of sentimentalism involved. He could smile at the idea of being in love with her. Yet some time he might be. He was no different from other men. She had made a profound impression on him. He knew that and did not attempt to shut his eyes to the truth. All these things, sinister and puzzling, of which her dead rider seemed the focus, might be of little consequence, after all. As far as he was concerned, every one simply insisted on taking him for a man who was dead. That had a comical aspect to Rock.
He stared with a speculative interest at the Parke ranch lying in the sunlight beside that shining river. Nona Parke had the right idea. She had the pick of a beautiful valley, eight hundred cattle, and the brains and equipment to handle them. That outfit would make a fortune for her and Betty. Yet it was a man’s job.
“She’s an up-and-coming little devil,” Rock said to himself. “Mind like a steel trap. Hard as nails. A man would never be anything more than an incident to her.”
Thus Rock unconsciously safeguarded his emotions against disaster. He was neither a fool nor a fish. He liked Nona Parke. He had liked her the moment he looked into the gray pools of her troubled eyes. But he wouldn’t like her too well. No; that would be unwise. She had warned him. But he could work for her. Her wages were as good as any—better, indeed, by ten dollars a month. And if there should be trouble in the offing—— Rock shrugged his shoulders. Bridge crossing in due time.
A moon-faced, dark-haired girl of sixteen was puttering around in the kitchen when Rock walked up to the house. Betty came flying to meet him, and Rock swung her to the ceiling two or three times, while she shrieked exultantly.
“Where’s Miss Parke?” he asked the half-breed girl.
“Workin’ in the garden.”
“Where the dickens is the garden?” Rock thought, but he didn’t ask. He went forth to see.
Ultimately he found it, by skirting the brushy bank of the river to the westward beyond the spring. Its overflow watered a plot of half an acre, fenced and cultivated. Rich black loam bore patches of vegetables, all the staple varieties, a few watermelon vines, and cornstalks as tall as a man. In the middle of this, Nona was on her knees, stripping green peas off a tangle of vines.
“Did Mary give you your dinner?” she asked.
“I struck the Maltese Cross round-up about eleven and ate with them,” he told her.
“Oh! Did you see Charlie Shaw?” she asked. “Did he say whether they picked up much of my stuff on Milk River?”
“Charlie Shaw is the name of that kid riding for you, eh? Well, I saw him, but he didn’t say much about cattle. And I didn’t ask. I had to step soft around that outfit. I don’t know any of these fellows, you see, and they all persist in taking me for Doc Martin. I suppose
I’d have a deuce of a time persuading anybody around here that I wasn’t.”
“It’s funny. I keep thinking of you as Doc, myself. You’re really quite different, I think,” she replied thoughtfully. “Somehow, I can’t think of Doc as being dead. Yet he is.”
“Very much so,” Rock answered dryly. “And I’m myself, alive, and I wish to stay so. I’ve been wondering if posing as your man, Doc, is, after all, a wise thing for me to do. What do you think?”
“You don’t have to,” she said quickly. “I’m sure Elmer Duffy would be relieved to know you aren’t Doc Martin.”
“I don’t know about that,” Rock mused. “Elmer might have just as much to brood over if he knew who I really am.”
“Why so?” she asked point-blank.
Rock didn’t question the impulse to tell her. His instinct to be himself was strong. The pose he had taken with Duffy that morning had arisen from mixed motives. He wasn’t sure he wanted to carry on along those lines. And he most assuredly didn’t want Nona Parke to think him actuated by any quixotic idea of functioning as her protector after her declarations on that subject.
So he told her concisely why Elmer Duffy might think a feud with Rock Holloway a sacred duty to a dead brother. Nona looked at him with wondering eyes and an expression on her face that troubled Rock, and finally moved him to protest.
“Hang it,” he said irritably. “You needn’t look as if I’d confessed to some diabolical murder. Mark Duffy was as hard as they make ’em. He was running it rough on an inoffensive little man who happens to be my friend. I had to interfere. And Mark knew I’d interfere. He brought it on himself. If I hadn’t killed him he would have killed me. That’s what he was looking for.”
“Oh, I wasn’t thinking that at all,” she said earnestly. “Of course, you were quite justified. I was just thinking that this explains why Elmer always hated Doc. Doc told me so. He felt it. I suppose it was the resemblance. I don’t see, now, so far as trouble with Elmer is concerned, that it matters much whether you pass as yourself or
Doc Martin. You’d have to watch out for Elmer Duffy in either case. I couldn’t trust that man as far as I could throw a bull by the tail.”
“Nice estimate of a man that’s in love with you,” Rock chuckled. “You’re a little bit afraid of Elmer, aren’t you?”
“No,” she declared. “But he’s brutal at heart. He’s the kind that broods on little things till they get big in his own mind. He would do anything he wanted if he was sure he could get away with it. And he would like to run both me and my ranch.”
“Powerful description,” Rock commented. “Still it sort of fits Elmer all the Duffys, more or less. They’re inclined to be more aggressive than they ought. Well, I guess it doesn’t make much difference if I do pass as Doc. I’m not trying to put anything over on anybody doing that. Now——”
He went on to tell her about meeting the girl at the Maltese Cross. He described the man who had glared at him and puzzled him by his attitude, but he didn’t tell Nona this latter detail. He merely wanted to know who was who.
“That was Buck Walters, range foreman of the Maltese Cross,” she confirmed Rock’s guess.
“Did Doc Martin ever have any sort of run-in with him?” he asked.
“Heavens, no! I would certainly have heard of it if he had. Why?”
“Oh, he seemed rather stand-offish, that’s all,” Rock answered indifferently.
“Buck thinks rather highly of himself,” Nona told him. “He’s in charge of a big outfit. The Maltese Cross is an estate, and he is one of the administrators. He’s pretty high-handed. There are men in this country who don’t like him much. But I don’t think Doc cared two whoops, one way or the other. Probably Buck was thinking about something.”
“Very likely. And who is the yellow-haired dulce?”
“Alice Snell. She and a brother inherit the whole Maltese Cross outfit when the boy comes of age.”
“She told me to tell you to ride down to see her—that she’d be at the ranch all summer.” Thus Rock delivered the message. “I didn’t hardly know what she was talking about.”
“Alice never does talk about anything much, although she talks a lot,” Nona said coolly. “Her long suit is getting lots of attention.”
“Well, I expect she gets it,” Rock ventured. “She’s good looking. Heiress to a fortune in cows. She ought to be popular.”
“She is,” Nona said—“especially with Buck Walters.”
“Oh! And is Buck popular with her?” Rock asked with more than mere curiosity. This was an item that might be useful in the task of sizing up Buck Walters and his way with the Maltese Cross.
“She detests him, so she says,” Nona murmured.
“Then why does she stick around up here in this forsaken country, when she doesn’t have to?”
“You might ask her,” Nona replied.
Rock had squatted on his heels, picking pods off the vines and chucking them by handfuls into the pan.
“I might, at that,” he agreed, “when I have a chance.”
“Alice is very ornamental,” Nona Parke continued thoughtfully. “But quite useless, except to look at. She gives me a pain sometimes, although I like her well enough.”
“You’re not very hard to look at yourself, it happens,” Rock told her deliberately. “And I don’t suppose you object to being ornamental as well as very useful and practical.”
Nona looked at him critically.
“Don’t be silly,” she warned.
“Don’t intend to be.” Rock grinned. “I never did take life very seriously. I sure don’t aspire to begin the minute I find myself working for you. I’m a poor but honest youth, with my way to make in the world. Is it silly for a man to admire a woman—any woman?”
“I wish you’d pull those weeds out of that lettuce patch,” she said, changing the subject abruptly. “They grow so quickly. I’m always at these infernal weeds. After you get that done, roll up your bed and bring it to the house. There’s lots of room.”
Rock performed the weeding in half an hour. If another had asked him to do that, he would probably have told him to go hire a
